math ia
Example 5: Student work
Maths Exploration
Newton-Raphson method
Rationale- For this project I chose to research and analyse the Newton-Raphson method, where calculus is used to approximate roots. I chose this topic because it looked extremely interesting and the idea of using calculus to approximate roots, seemed intriguing.
The aim of this exploration is to find out how to use the Newton-Raphson method, and in what situations this method is used
Explanation of the Newton-Raphson method
The Newton-Raphson or Newton’s method is an iterative process to approximate roots.
We know simple roots for rational numbers such as numbers such as
3 or
4 or 9 , but what about irrational
5 . This method was discovered in 1736 by Isaac Newton after
being published in the ‘Method of Fluxions’, this method was also described by Joseph
Raphson in 1690 in ‘Analysis Aequationum’.
The Newton-Raphson Process:
In the Newton-Raphson process the following formula is used:
Mathematics SL and HL teacher support material
1
Example 5: Student work
xn += xn −
1
f ( xn )
,
f '( xn )
f '( xn ) ≠ 0
This is called the Newton-Raphson formula
How this formula was derived:
The square root of a number (n) can be found by using the function:
f ( x= x 2 − n
)
The root of n is the value of x when f(x) = 0
The first thing you do when approximating a root is to make an initial estimate in terms of a positive integer and find the tangent of the function at that point.
Let the initial estimate = x₁
The intercept of the tangent at x₁ with the x-axis will be closer to the root than the initial estimate. This intercept can be called x₂.
Calculating x₂ is done by studying the triangle bounded by the x-axis, the line x=x₁, and the tangent to the line at x₁. This only works when the gradient of the function at x₁ is not equal to zero.
An example of a triangle is shown in the following graph (where the square root of 7 is being estimated):
Maths Exploration
Newton-Raphson method
Rationale- For this project I chose to research and analyse the Newton-Raphson method, where calculus is used to approximate roots. I chose this topic because it looked extremely interesting and the idea of using calculus to approximate roots, seemed intriguing.
The aim of this exploration is to find out how to use the Newton-Raphson method, and in what situations this method is used
Explanation of the Newton-Raphson method
The Newton-Raphson or Newton’s method is an iterative process to approximate roots.
We know simple roots for rational numbers such as numbers such as
3 or
4 or 9 , but what about irrational
5 . This method was discovered in 1736 by Isaac Newton after
being published in the ‘Method of Fluxions’, this method was also described by Joseph
Raphson in 1690 in ‘Analysis Aequationum’.
The Newton-Raphson Process:
In the Newton-Raphson process the following formula is used:
Mathematics SL and HL teacher support material
1
Example 5: Student work
xn += xn −
1
f ( xn )
,
f '( xn )
f '( xn ) ≠ 0
This is called the Newton-Raphson formula
How this formula was derived:
The square root of a number (n) can be found by using the function:
f ( x= x 2 − n
)
The root of n is the value of x when f(x) = 0
The first thing you do when approximating a root is to make an initial estimate in terms of a positive integer and find the tangent of the function at that point.
Let the initial estimate = x₁
The intercept of the tangent at x₁ with the x-axis will be closer to the root than the initial estimate. This intercept can be called x₂.
Calculating x₂ is done by studying the triangle bounded by the x-axis, the line x=x₁, and the tangent to the line at x₁. This only works when the gradient of the function at x₁ is not equal to zero.
An example of a triangle is shown in the following graph (where the square root of 7 is being estimated):